Atmospheric Science 4310 7310

1
Atmospheric Science 4310 / 7310

• Atmospheric Thermodynamics
• By
• Anthony R. Lupo

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Syllabus

• Atmospheric Thermodynamics
• ATMS 4310
• MTWR 900 950 / 4 credit hrs.
• Location 1-120 Agruculture Building
• Class Ref 15505
• Instructor A.R. Lupo
• Address 302 E ABNR Building
• Phone [phone]
• Fax [phone]
• Email lupo_at_bergeron.snr.missouri.edu or
  LupoA_at_missouri.edu
• Homepage www.missouri.edu/lupoa/author.html
• Class Homepage www.missouri.edu/lupoa/atms4310.h
  tml
• Office hours MTWR 1000 1050
• 302 E ABNR Building

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Syllabus

• Grading Policy Straight
• 97 100 A 77 79 C
• 92 97 A 72 77 C
• 89 92 A- 69 72 C-
• 87 89 B 67 69 D
• 82 87 B 62 67 D
• 79 82 B- 60 62 D-
• Grading Distribution
• Final Exam 20
• 2 Tests 40
• Homework/Labs 35
• Class participation 5 (Note, you WILL lose 1
  point for each unexcused absence, up to 5 points.
  This IS a half-letter grade, keep that in mind!)
• Attendance Policy Shouldnt be an issue!

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Syllabus

• Texts
• Holton, J.R., 2004 An Introduction to Dynamic
  Meteorology, 4th Inter, 535 pp. (Required)
• Bluestein, H.B., 1992 Synoptic-Dynamic
  Meteorology in the Mid-latitudes Vol I Priciples
  of Kinematics and Dynamics. Oxford University
  Press, 431 pp.
• Hess, S.L., 1959 An Introduction to Theoretical
  Meteorology. Robert E. Kreiger Publishing Co.,
  Inc., 362 pp.
• Zdunkowski, W., and A. Bott, 2003 Dynamics of
  the Atmosphere A course in Theoretical
• Meteorology. Cambridge University Press, 719 pp.
  (a good math review)
• Zdunkowski, W., and A. Bott, 2004 Thermodynamics
  of the Atmosphere A course in Theoretical
  Meteorology. Cambridge University Press, 251 pp.
• Various relevant articles from AMS and RMS
  Journals.
• Course Prerequisites
• Atmospheric Science 1050, Calculus through Math
  1700, Physics 2750, or their equivalents. Senior
  standing or the permission of the Instructor.

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Syllabus

• Calendar Wednesday is Lab exercise day
• Week 1 21 22 23 24 August ? Introduction and
  Friday makeup arrangements. Intro. To Atms. 4310.
  Lab 1 The Thermodynamic diagram and upper air
  information.
• Week 2 28 29 30 31 August / September ? Fri.
  makeup 1, 1 September. Lab 2 Adiabatic Motions
  in the Atmosphere.
• Week 3 hh 5 6 7 September ? Mon., Labour Day
  Holiday / Fri makeup 2, 8 September. Lab 3 The
  Thermodynamic Diagram Examining Moist Processes.
• Week 4 11 12 13 14 September ? Fri. makeup 3,
  15 September, Lab 5 Lab 4 The Thickness Equation
  and its Uses in Operational Meteorology. (move
  up other labs)
• Week 5 18 19 20 21 September ? Friday make up
  4, 22 September, Lab 5 the Lapse Rates of Special
  Atmospheres. Test 1 22 Sept., covering material
  to 19 Sept.?
• Week 6 25 26 27 28 September ? Friday 29
  September makeup 5.
• Lab 6 Using Thermodynamic diagrams to Determine
  Water Vapor Variables.
• Week 7 2 3 4 5 October ? Friday makeup 6, 6
  October. Lab 7 Estimating Vertical Motions Using
  the First Law of Thermodynamics.

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Syllabus

• Week 8 nn nn nn nn October ? No Class, UCAR-NCAR
  member rep meetings and Heads and Chairs. Lab 8
  Atmospheric Stability I Special Forecasting
  Problems Fog Formation.
• Week 9 nn nn nn nn October ? Gone to Cleveland,
  OH NWA meet.
• Lab 9 Atmospheric Stability II Special
  Forecasting Problems Air Pollution.
• Week 10 23 24 25 26 October ? Makeup 7, 27
  October
• Lab 10 Severe Weather The Synoptic-Scale sets
  the table.
• Week 11 30 31 1 2 October / November ? Makeup
  8, 3 November Test covering material to 1
  November. Lab 11 Using Thermodynamic diagrams in
  forecasting Convective Outbreaks.
• Week 12 nn nn nn nn November ? Severe and Local
  Storms Conference in Saint, Louis, MO. Lab 12
  Estimating Various Stability Indicies in
  real-time.
• Week 13 13 14 15 16 November ? Makeup number 9,
  17 November Lab 13 Severe Weather I Using
  thermodynamic diagrams Super Cell Formation and
  Wind Gust Estimation.
• Week 14 hh hh hh hh November ? No classes Turkey
  day week!
• Week 15 27 28 29 30 November / December ? Make
  up number 10, 1 December. Lab 14 Severe Weather
  II Using thermodynamic diagrams Hail Formation
• Week 16 4 5 6 7 December ? Makeup 11, 8 Dec.,
  Final 8 Dec.?

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Syllabus

• ATMS 4310 Final Exam
• The Exam will be quasi-comprehensive. Most of the
  material will come from the final third of the
  course, however, important concepts (which I will
  explicitly identify) will be tested. All tests
  and the final exam will use materials from the
  Lab excercises! Thus, all material is fair game!
  The final date and time is
• Friday, 15 December 2006 1030 am to 1230 pm
  in ABNR 1-120
• University Important Dates Calendar
• August 14-18 FS2006 Regular Registration
• August 16 Residence Halls open 900 a.m.
• August 18 Easy Access registration - noon - 600
  p.m.
• August 21 Classwork begins 800 a.m.
• August 21 Late Registration and Add/Drop - Late
  fee assessed
• beginning August 21
• August 28 Last day to register, add, or change
  sections
• August 29-Sept. 25 Drop Only
• September 4 Labor day Holiday
• September 5 Last day to change grading option
• September 18 (Census Day) - Last day to register
  for CDIS courses
• for Fall

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Syllabus

• TBA WS2007 Early Registration Appointments
• October 30 Last day to withdraw from a course -
  FS2006
• November 15 Last day to change divisions
• November 18 Thanksgiving recess begins, close of
  day
• November 27 Classwork resumes, 800 a.m.
• December 8 Fall semester classwork ends
• December 8 Last day to withdraw from University
• December 9 Reading Day
• December 11 Final examinations begin
• December 15 Fall semester ends at close of day
• December 15-16 Commencement Weekend
• Please note This calendar is subject to change

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Syllabus

• Syllabus
• Introductory and Background Material, including a
  math review (Calculus III)
• The Thermodynamics of Dry Air
• Hydrostatics
• The Thermodynamics of Moist Air
• Static Stability and Convection
• Vertical Stability, Instability, and Convection
• Cloud Microphysics
• The Thunderstorm and Non-hydrostatic Pressure
• These topics will be taught if there is time.
  All Lecture schedules are tentative!

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Syllabus

• Special Statements
• ADA Statement (reference MU sample statement)
• Please do not hesitate to talk to me!
• If you need accommodations because of a
  disability, if you have emergency medical
  information to share with me, or if you need
  special arrangements in case the building must be
  evacuated, please inform me immediately. Please
  see me privately after class, or at my office.
• Office location 302 E ABNR Building Office
  hours ________________
• To request academic accommodations (for example,
  a notetaker), students must also register with
  Disability Services, AO38 Brady Commons,
  882-4696. It is the campus office responsible for
  reviewing documentation provided by students
  requesting academic accommodations, and for
  accommodations planning in cooperation with
  students and instructors, as needed and
  consistent with course requirements. Another
  resource, MU's Adaptive Computing Technology
  Center, 884-2828, is available to provide
  computing assistance to students with
  disabilities.
• Academic Dishonesty (Reference MU sample
  statement and policy guidelines)
• Any student who commits an act of academic
  dishonesty is subject to disciplinary action.

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Syllabus

• The procedures for disciplinary action will be in
  accordance with the rules and regulations of the
  University governing disciplinary action.
• Academic honesty is fundamental to the activities
  and principles of a university. All members of
  the academic community must be confident that
  each person's work has been responsibly and
  honorably required, developed, and presented. Any
  effort to gain an advantage not given to all
  students is dishonest whether or not the effort
  is successful. The academic community regards
  academic dishonesty as an extremely serious
  matter, with serious consequences that range from
  probation to expulsion. When in doubt about
  plagiarism, paraphrasing, quoting, or
  collaboration, consult the instructor. In cases
  of suspected plagiarism, the instructor is
  required to inform the provost. The instructor
  does not have discretion in deciding whether to
  do so.
• It is the duty of any instructor who is aware of
  an incident of academic dishonesty in his/her
  course to report the incident to the provost and
  to inform his/her own department chairperson of
  the incident. Such report should be made as soon
  as possible and should contain a detailed account
  of the incident (with supporting evidence if
  appropriate) and indicate any action taken by the
  instructor with regard to the student's grade.
  The instructor may include an opinion of the
  seriousness of the incident and whether or not
  he/she considers disciplinary action to be
  appropriate. The decision as to whether
  disciplinary proceedings are instituted is made
  by the provost. It is the duty of the provost to
  report the disposition of such cases to the
  instructor concerned.

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Syllabus

• Lab Exercise Write-up Format All lab write-ups
  are due at the beginning of the next lab
  Wednesday. Grading format also given.
• Total of 100 pts Name
• Lab
• Atms 4310
• Neatness and Grammar 10 pts Date Due
• Title
• Introduction brief discussion of relevant
  background material (5 pts)
• Purpose brief discussion of why performed (5
  pts)
• Data used brief discussion of data used if
  relevant (5 pts)
• Procedure (15 pts)
• 1.
• 2.

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Syllabus

• Results brief discussion of results (50 pts)
• observations
• discussion (answer all relevant questions here)
• Summary and Conclusions (10 pts)
• summary
• conclusions
• Write-ups need to be the appropriate length for
  the exercise done. If one section does not apply,
  just say so. However, one should never exceed 6
  pages for a particular write up. Thats too
  much! Finally, answer all questions given in the
  assignment.

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Day 1

• Thermodynamics ? the study of initial and final
  equilibrium states of a "system" which has been
  subjected to a specified energy process or
  transformation.
• System ? a specific sample of matter (air
  parcels)
• We will concentrate on the macroscale or parcel
  properties only! We will not look at the
  microscale (molecular level) thats atmospheric
  physics (Dr. George, Dr. Fox).

15
Day 1
16
Day 1

• Variables of state (thermodynamic variables)
• pressure (hPa, mb) (Force Area-1)
• Temperature (oC, K, oF)
• Volume (typically m3 kg-1), but typically
  assume unit mass)

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Day 1

• Laws of thermodynamics
• Equation of State
• 1st law of thermodynamics (conservation of
  energy)
• 2nd law of thermodynamics (entropy) (direction of
  heat flow) (warm to cold)
• We will review Dimensions and Units, and
  conventions.

18
Day 1

• Atmospheric science derives a set of standard
  measurements or unit system, such that everyone
  everywhere will be on the same page.
• AMS endorsed the SI (Systeme International) or
  International system of Units (BAMS, 1974, Aug.)
• The basic units are Length, Mass, Time (meter,
  m kilo, kg second, s)

19
Day 1

• A derived unit combines basic units example,
  pressure
• Force /Area kg m s-2 / m2 kg m-1 s-2
  Pascals
• 1000 Pa 1 kPa 10 hPa 10 mb
• Temperature (Kelvin, or absolute scale Celsius
  (1742) Farenheit (1714)).
• Coordinate System (Cartesian)

20
Day 1

• Coordinate system tangent to Earths surface
  which is really a sphere (curvature for most
  applications and approximations can be
  neglected).
• Cartesian coordinates
• x,y,z,t x,y,p,t, or x,y,q,t
• Could also use natural coordinates
• s(treamline),n(normal),z,t
• Spherical coordinates
• r(adius),q(longitude),f(latitude)

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Day 2

• Wind
• Wind direction direction from which the wind
  blows, and compass direction, not Cartesian!
• West wind is blowing from 270o.
• Wind direction increasing with time or height
  veering decreasing with time or height backing

22
Day 2

• Remember Direction of math 270 compass
  (meteorology) direction.
• Vector representation in Geophysical Fluid
  Dynamics
• Remember the atmosphere is a fluid, and a fluid
  is liquid or gas. Thus, the primitive equations
  will be valid in any atmosphere, terrestrial
  (extraterrestrial).

23
Day 2

• Scalar quantity ? A quantity with magnitude only
  (e.g., wind
• speed has units m s-1) (zero order tensor)
• Vector ? (first order Tensor) A quantity with
  magnitude and direction (e.g., wind velocity)

24
Day 2

• Wind

25
Day 2

• Dyadic? (2nd order Tensor) has a magnitude and
  two directions!
• Example stress (Force per unit area), where A is
  the vector of some magnitude equal to the area
  and in the direction of the normal.
• In English Magnitude, direction (1) of the
  force, and (2) on which surface applied

26
Day 2

• An example
• Example Stress Force Area times Stress (has
  same units as pressure)

27
Day 2

• Vector Analysis
• Vector Notations
• A A
• A magnitude of A

28
Day 2

• Vectors are equal if they have equal magnitude
  and directions!!
• The unit vector any vector of unit length!
• where is a vector of unit
  length
• Cartesian Unit vectors vectors of
  unit length in the positive x,y,z direction,
  respectively.

29
Day 2

• Natural coordinates are
• Vector components (2 dimensions), but we can
  extend to infinite number of directions
• Magnitude of A ? A ? (Ax2 Ay2)

30
Day 2

• Vector addition and subtraction
• 1) A B C
• 2) AB B A C
• 3) A - B C

31
Day 2

• Addition Subtraction

32
Day 2

• Heres how
• Associative rule
• (AB) C A (BC)
• Negative Vector Is a vector of the same
  magnitude, but opposite direction.

33
Day 2

• Vector multiplication
• Scalar x Vector
• In the atmospheric sciences The wind vector (2-D
  3 D)

34
Day 2 /3

• Vector products
• The dot product (also the scalar or inner)
  product
• A dot B ABcos(q)
• Physically The dot product is the PROJECTION of
• vector B onto Vector A in direction of A!

35
Day 2/3

• Projection (mathworld.wolfram.com) (excellent
  math site)

36
Day 3

• Properties of the Dot Product
• commutative
• A dot B B dot A
• associative
• A dot (B dot C) (A dot B) dot C
• distributive
• A dot (B C) A dot B A dot C

37
Day 3

• Dot product of perpendicular vector 0
• In order for the dot product to have a value, the
  B vector must have a component parallel to vector
  A!
• Recall cos(0o) 1 and cos(90o) 0
• Thus, i dot j, and j dot k, etc 0, and i
  dot i 1, etc

38
Day 3

• Orthogonal Vectors (Orthogonality property) When
  the angle between two vectors is 90o, or the dot
  product is zero, two vectors are said to be
  orthogonal.
• Other Dot Product Rules
• 1) A dot A AA cos(0) A2
• 2) A dot mA mA2

39
Day 3

• Dot product of two vectors (heres how)
• Remember Foil?
• AxBx (i dot i) Ax By ( i dot j) Ay Bx (j dot
  i) Ay By (j dot j)

40
Day 3

• Ok, now you try ?
• Answer????? Ax
• Cross Product (or vector product)
• A x B ABsin(q)

41
Day 3

• What is it (again courtesy of Mathworld site)?

42
Day 3

• Einstien notation permutation
• Q "What do you get when you cross a
  mountain-climber with a mosquito?"
• A "Nothing you can't cross a scaler with a
  vector,"
• Q "What do you get when you cross an elephant
  and a grape?"
• A "Elephant grape sine-of-theta."

43
Day 3

• The cross product of two vectors is a third
  vector that is mutually perpendicular to the two
  vectors and the plane containing these vectors.
• Q Remember your Physics?
• The positive direction of A x B may be determined
  by the right hand (or corkscrew) rule. Just
  curl your fingers from A to B, and your right
  thumb (vector C) is the result! (ayyyy)

44
Day 3

• Evaluating the cross product of A and B in the
  cartesian coordinate system.
• The vector or cross product is NOT commutative!

45
Day 3

• Try the right hand rule to show that we cannot
  switch order
• Also, remember sin(-90o) -1.0
• The cross product is distributive
• A X (B C) (A x B) (A x C)

46
Day 3/4

• The cross product of a vector with itself equals
  0!!
• q 0 so, sin(0) 0, or A x A
  AAsin(0) 0
• Cross products of unit vectors and the cyclical
  property of the cross product
• 1) i x i 0, i x j k, i x k -j
• 2) j x i -k, j x j 0, j x k i
• 3) k x i j, k x j -i, k x k 0

47
Day 4

• A x B C , B x C A, C x A B
• Another trick or property
• Unit vector k x Ah Horizontal vector of length
  A turned 90o to the left of A (try it with
  right hand rule - horizontal vector!!!)
• -k x Ah A vector of length A turned 90o to the
  right of A

48
Day 4

• Multiple Vector Products
• Scalar Triple product
• A dot (B x C) a scalar value
• This is also cyclical
• A dot (Bx C) B dot (C x A) C dot (A x B)

49
Day 4

• Or dot and cross product may be interchanged
• A dot ( B x C) (A x B) dot C
• Triple vector product
• (A x (B xC)) Vector quantity
• A x (B xC) (A dot C ) B (A dot B) C
• The result is a third vector in the plane of B
  and C!!!

50
Day 4

• But,
• (A x B) x C not equal to A x (B x C)
• since for former result is in the plane of A and
  B!!

51
Day 4/5

• The mathematical description of the Atmosphere
• We must eventually develop from fundamental
  physical laws and concepts (first principles),
  the 5 vector (7 scalar) equations of geophysical
  fluid dynamics, and describe and understand the
  behavior of the atmosphere through the
  manipulation of these equations.
• Describing the atmosphere in terms of the
  distribution in space and time of certain
  properties of the atmosphere.

52
Day 5

• Independent variables ? these are the basis of
  (or describe) our coordinate system.
• Well use Cartesian system (x,y,z,t)
• and a right handed coordinate system

53
Day 5

• Dependent variables ? depend on your position in
  space and time, and can be described as a
  function of the independent variables.
• In atmospheric science u,v,T,r,q,P,w,or w
• Example u(x,y,z,t)

54
Day 5

• Invariance ? A quantity that does not change if
  measured in a different coordinate system
• e.g., rotationally invariant ? quantities
  that do not change even if coordinate system
  rotates
• Q Which variable might be rotationally
  invariant?
• A T(x,y,z,t) T(x,yz,t)
• Galilean invariant ? quantities that do not
  change even if coordinate system is moving
  horizontally

55
Day 5

• Important Definintion!
• Conserved ? a quantity that does not change with
  time. (e.g., Potential Temperature and adiabatic
  motions)
• Conservation ? the change in some quantity with
  time equals 0!

56
Day 5

• Conservation steady state balance (between
  sources and sinks!)
• where Q any quantity

57
Day 5

• The derivative (A review)
• Let take a quantity Q(x,y,z,t)
• Now one needs to take the Total Derivative. In
  order to do this, we must use the Chain Rule!
  Remember this?

58
Day 5

• Total derivative is (in x,y,z,t,)
• advective derivative Eulerian
• (in x,y,p,t)

59
Day 5

• in (x,y,q,t)
• in natural coordinates

60
Day 5

• now lets give our pens a break
• 1) u dx /dt,
• 2) v dy / dt,
• 3) w dz / dt
• In Mathematics
• The total derivative (heavy D) (substantial,
  individual, material) is exact, thus the
  derivative not path dependent!

61
Day 5

• Exactness!

62
Day 5

• But, if path dependent, total derivative has no
  meaning, and we write with a small d.
• If path dependent, then the process is sensitive
  to the initial starting place and it corresponds
  to (generally) one outcome.
• But I, like most atmospheric scientists use the
  notation d for a total derivative regardless.

63
Day 5

• The partial derivative is
• - the change in one variable or coordinate w/out
  regard to the other components.
• looks like..
• in the eyes of the partial derivative here, where
  C z2y2t2.

64
Day 5/6

• Differentiation of Vectors
• The normal rules of differentiation apply, but
  you must preserve the order when cross product is
  applied.
• Let vector A be time dependent, i.e., vector A is
  changing size and/or direction with time and
  space

65
Day 6

• If our coordinate system is not changing (i.e.,
  i,j,k constant).
• then Ax,Ay,Az change (of course!)
• (oh, you fill in the rest!)

66
Day 6

• Some more fun rules
• 1)
• 2)
• 3)

67
Day 6

• but. if i,j,k are changing.
• Position, Velocity, Acceleration
• The position vector R, the velocity vector V and
  the Acceleration vector A

68
Day 6

• Position vector
• The velocity vector
• The acceleration vector

69
Day 6

• The del operator
• Also known as the Hamiltonian, gradient, or
  nabla operator.
• Let us define a differential operator with vector
  properties

70
Day 6

• The del operator has no physical meaning until it
  operates on another quantity such as a scalar or
  another vector! (A ghost vector)
• Operating on a scalar

71
Day 6

• Del Q is now a 3-D vector whose direction is in
  the direction of the maximum increase of Q and
  whose magnitude is equal to the rate of change of
  Q per unit distance in that direction.

72
Day 6

• Del Q in normal (perpendicular to lines of Q). On
  a 2 D surface is perpendicular to Q isolines.
• Then in plane English delQ is simply the slope
  of Q on some planar surface. The first derivative
  in space (slope) is analogous to the first
  derivative in time (velocity).

73
Day 6/7

• Now a proof! (show velocity vector is
  perpendicular to gradient vector)
• Step 1
• But, dQ 0 on a line of Q (correct?)

74
Day 7

• Now we know that
• a) each point of a surface of constant Q can be
  defined by the position vector.
• b) then on a Q surface dr (or V), must be on the
  surface of Q, so dr must lie on the Q surface.

75
Day 7

• c) then dr dot del Q on Q surface
• So dr dot delQ dQ, this is the definition of
  the total derivative.
• but dQ 0 on Q surface as discussed above.

76
Day 7

• Therefore since dr and delQ separately are not 0,
  but their dot product IS 0, they must be
  perpendicular (or orthogonal)!!!
• Furthermore, we know that delQ was perpendicular
  to lines of Q, thus dr or Velocity, must be
  parallel to lines of Q!
• Point proved!

77
Day 7

• Advection of a scalar quantity
• 3-D transport of some quantity
• In Atms Sci, in (x,y,p,t) coordinates, advection
  and flux are equivalent.

78
Day 7

• A 2-D example of advection
• Some other names for advective quantity
• Convective derivative or Lagrangian!

79
Day 7

• Two definitions
• Lagrangian measurement ? measurement that moves
  with the flow (goes with the flow)
• Eulerian ? measurement at a stationary point
• Important Concept!!
• ? If Q is conserved or a conserved property
  (invariant with time) time derivative 0. We
  also call this steady state.

80
Day 7

• Thus, if this is true either the source-sinks are
  zero, or Equal and opposite. But it also implies
  that advection equals the time rate of change.

81
Day 7

• - or
• where Q Any variable, vector or scalar!

82
Day 7

• The Laplacian operator
• ? Del dot Del Its a scalar operator! It
  typically changes the sign of a function. A
  measure of the curvature in a function.
• ? as acceleration is to velocity, so is the
  gradient operator (slope) to the Laplacian
  (curvature)!

83
Day 7

• The divergence of a vector (del dot V) (a
  scalar)!
• Velocity divergence
• ? ?

84
??????? ???

• The curl of the vector V
• The curl of the velocity or vector vorticity

85
Day 7

• Vertical component of Vorticity (zeta)
• Magnitude in vertical is entirely dependent on
  horizontal spatial variations or shears
• Consider a case where all the terms contribute
  positively

86
Day 7/8

• Then
• ? Then, the vertical component is POSITIVE for
  cyclonic circulations (shear) and negative for
  anticylonic circulations. It is the opposite in
  the SH.

87
Day 8

• Rotational Vectors and vectors in Rotation
• Definition of the rotational vector (omega - w) ?
  rotational vector has a direction along the axis,
  positive in the sense of the Right hand rule, and
  its magnitude, omega w, is porportional to
  the angular velocity of the rotating system.
  (ang. Vel. radians/sec)
• The rate of change of a vector A of constant
  magnitude due to its changing direction produced
  by rotation (omega - W)

88
Day 8

• Now look down from above at the plane in which
  the vector A is rotating.

89
Day 8

• The magn. Or length of DA DA and for small
  angle Dq
• Recall from Geometery
• DA A sin(f) Dq
• ? since for small angles tan q q opp. / adj.
  Or opp. adj. tan q

90
Day 8

• So, (now include delta t)
• DA/ Dt A sin(f) (Dq / D t)
• And as Dt goes to 0 ?
• dA/dt A sin(f)(dq /dt)

91
Day 8

• But dq /dt omega w (the angular velocity) ? So
• dA/dt w A sin (f)
• Then we need to
• ? redefine w as W since were talking about
  earth!
• ? recall our definition of the cross product!!!
  (W x A)

92
Day 7

• Thus the magnitude of DA/dt DA/dt equals the
  mangitudes of W x A!!
• dA/dt WxA!

93
Day 8

• Direction of dA/dt and W x A
• ? Since (W x A is mutually perpendicular to W and
  A as is DA/dt, and is positive in the same
  direction the two vectors DA/dt and W x A are
  equal vectors!
• Thus for A of constant magnitude dA/dt W x A

94
Day 8

• ? The rate of change of vector A in a fixed
  (absolute) coordinate system vs. a rotating
  coordinate system. (Fixed and absolute are
  not good news we cant do this in practice (in
  real atms.)).
• Atms example coriolis force!!
• dV/dt W x V

95
Day 8

• OK, theres more than one way to skin a cat ..
• Consider (X,Y,Z) w/ unit vectors (I,J,K)
• Consider another (x,y,z) w/unit vectors (i,j,k).
  Allow this one to rotate w/angluar velocity
  (omega).
• A AXI AYJ AZK Axi Ayj Azk

96
Day 8

• Now differentiate A w/r/t time (sytem 1 is
  const.! System 2 is rotating)
• DA/dt AX/dt I dAY/dt J dAZ/dt K dAx/dt i
  Ax di/dt etc..
• i,j,k are unit vectors.
• Di/dt W x i and dj/dt W x j and.

97
Day 8

• (DA/dt)abs dAx/dt i dAy/dt j dAz/dt k (Ax
  (W x i) Ay (W x j) Az (W x k) )
• DA/dt abs (dA/dt) (relative to rotating coord.
  System) (W x (Axi Ayj Azk)
• DA/dt dA/dt (relative to rot.) W x A (rot of
  coord system w/r/t vertical)

98
Day 8/9

• The rate of change of a vector in an absolute
  (inertial) frame of ref. Is equal to rate of
  chage observed in the rotating system a term
  to the cross product of the rotational vector and
  the arbitrary vector A!
• ? Thus, you have just derived the expression for
  the coriolos force! ?

99
Day 9

• Dimensional Analysis (The rules!)
• 1. All terms of an equation must have the same
  dimensions! e.g. potential temp relationship
• 2) All exponents are non dimensional
• 3) All log and trig functions are also
  non-dimentional.

100
Day 9

• 4. The dimensions of differentials are the
  same as the dimensions of a differentiated
  quantity.
• e.g., we can say d (ln (T))
• ? Note specific as a prefix implies the
  quantity is per unit mass and has the dimensions
  of (Q x M-1) e.g., specific volume vol / unit
  mass.

101
Day 9

• ? Valid physical relationships must be
  dimensionally consistent with eachother, in other
  words X Y must have the same units. Otherwise,
  we say X proportional to Y. Or X AY where the
  units of AY are the same as X.
• Caveat Some proportionalities have consistent
  units.

102
Day 9

• Non-dimensional analysis (Scale analysis)
• ? (Mathematical formalization) Theoreticians like
  to look at equations in a non-dimensional sense,
  that is we choose characteristic time and space
  scales for some phenomena to be studied.
• We essentially change coordinate systems
• (x,y) L(x,y) where L is 2000 km
• t Tt where T 100000 sec
• (u,v) U(u,v) U 10 m/s

103
Day 9

• ? In performing this type of analysis we can
  determine what processes are important for du/dt,
  for example, in the horizontal equation of motion
  on the desired scale (cyclone), we can perform
  this type of analysis for particular phenomena.
• Informal Scale analysis
• ? Similar to that of non-dimensionalization.
  Scale (size) analysis is also a powerful tool
  often used in meteorological derivations the
  analysis of physical processes. This is a more
  informal method than non-dimensional analysis.

104
Day 9

• ? Suppose we have an equation based on a
  physical law or principle (e.g. Newtons 2nd law,
  3rd eqn. of motion). This equation is a
  generalized equation, valid for many atmospheric
  phenomena.

105
Day 9

• We choose the scale we are interested in, the
  consider the order of magnitude, (e.g. the size
  and space scale) each term in the equation would
  have, for that particular scale of motion (again
  typical values, or estimates).
• Then, we can neglect the smaller terms and
  simplify the equation. Well also know the error
  introduced in doing this (ratio of neglected to
  retained terms).
• The equation is simplified, but now less general,
  its the trade-off for a simpler relationship.

106
Day 9

• Hydrostatic balance
• This equation governs the movement of
  synoptic-scale systems (highs/lows (waves in the
  westerlies)) and fronts.

107
Day 9

• Hydrostatic balance the error ?
• Error Neglected Terms / Retained terms
• Error 10-3 / 10 10-4 0.0001 0.01
• ? Thus this is a darned good estimate for
  synoptic and meso alpha scales.
• ? Important Point! The equation is much simpler,
  but its only valid for these scales and has now
  lost its generality!

108
Day 9/10

• Scales of Atmospheric motions
• Scale Horiz Dimension Time
• Planetary 10,000 km weeks - 1 month
• Synoptic 2,000 6000 km 1 to 7 days
• Meso 10 km 2,000 km 1h 1 day
• Micro
• ? Can you think of examples of real phenomena
  that fit each category?

109
Day 10

• Fundamental equations of geophysical
  hydrodynamics
• ? Seven dependent variables, four independent
  variables, seven independent equations
• p(x,y,z,t) Pressure (mass)
• r(x,y,z,t) density (mass, thermal)
• T(x,y,z,t) or q(x,y,z,t) (potential) Temperature
  (thermal)
• M(x,y,z,t) Mixing Ratio (mass, thermal)
• U(x,y,z,t) zonal wind (mass)
• V(x,y,z,t) meridional wind (mass)
• W(x,y,z,t) vertical wind (mass)

110
Day 10

• Concept Name
• Elemental Kinetic Theory Eqn. Of state

111
Day 10

• Cons. of Energy 1st Law of Thermo.
• Cons. of Mass Eq. of continuity

112
Day 10

• Cons. of mass Eq. of water mass

113
Day 10

• Cons. of momentum Eq. of motion
• A.K.A Navier Stokes Equation, Newtons 2nd
  Law, etc.

114
Day 10

• Von Helmholtz 1858 In principle this is a
  mathematically solvable system (closed) given
  observed initial state and proper BCs, the
  solution should yield all future states of the
  system. (The rub ICs and BCs).
• ? Thus, forecasting is an initial value problem
  (Bjerknes, 1903)
• ? These eqns. Are what will be studied in Atms
  4310, 4320. These describe behavior of the
  atmosphere.
• ? Solving these equations (Numerical methods and
  modeling classes Atms 4800)

115
Day 10

• The Thermodynamics of Dry Air (Holton Ch 2- 4)
• Reminder Dry air means DRY air (no moisture)!
• Moist air means water vapor present.
• Dry air (a homogeneous mixture of gasses from 0
  80 km up Homosphere)

116
Day 10/11

• The Heterosphere is above that, gasses separate
  by weight (mass)
• The Atmosphere its makeup
• Gas (atomic weight) by Vol by mass
• Nitrogen (N2) 28.02 78.1 75.5
• Oxygen (O2) 32.00 20.9 23.1
• Argon (Ar) 39.94 0.93 1.3
• Carbon Dioxide (CO2) 44.01 0.036 0.05

117
Day 10/11

• ? Many other gases present in very small
  quantities (Ne, He, H, O3) they are called Trace
  Gases
• ? Thus if we calculate the atomic weight of air
• 28.97 kg mol-1
• ? Three of these are very important because
  despite the small quantities, they help determine
  the temperature structure of the troposphere and
  stratosphere H2O, CO2, and O3

118
Day 11

• H2O is important within the hydrologic cycle,
  clouds, rain etc.. Water is the only substance in
  earth atmosphere that exists in all three phase
  at terrestrial pressures and temperatures.
• ? Water Vapor and clouds are important in
  determining atmospheric structure due to their
  radiative properties (albedo, infrared).
• ? Residence time 1 10 days.

119
Day 11

• ? It is the most important and potent greenhouse
  gas, but its not homogeneously distributed!
• CO2 has homogeneous concentration. It is
  important because of its radiative properties in
  infrared. Its residence time near 100 years, thus
  important in longer term climate change. But, the
  CO2 cycle is not well known yet.
• O3 concentrated at 32km up (in stratosphere)
  due to solar and chemical reactions. It absorbs
  UV and emits infrared and responsible for the
  Stratospheres inversion.

120
Day 11

• ? Near surface it is present in small amounts due
  to pollution, but it is highly poisonous.
• Moist air
• ? Water vapor extremely variable near 0 near 4
  
• Thats 0 - 40 g/kg!
• More typical 1 10 g/kg (Td 57F)

121
Day 11

• The variables of state
• Mass (M)
• ? Density (mass/unit vol) (r)
• ? Specific volume (vol / unit mass) a
• ? Pressure (Force/Area) is due to molecular
  collisions and the associated momentum changes
  independent of direction (a scalar). The force is
  normal to gas container walls. N m-2 1 Pa
  and 1mb 102 Pa - or - kg m-1s-1

122
Day 11

• ? Temperature (T) a measure of the average
  internal energy of the molecules obtained during
  a state of equilibrium.
• ? To read temperature there needs to be
  equilibrium established between the system and a
  temp sensor. Temp. determines direction of heat
  flow. (Kelvin, Celsius) 0o C 273.15 K.
• Ideal Gas (Kinetic theory of gasses) is a
  collection of molecules that are
• completely elastic spheres
• with no attractive or repulsive forces, and
• occupying no volume.

123
Day 11

• Heat is a form of energy, which can be
  transferred from a warmer to colder aubstance.
  Heat transfer by (really kinetic energy
  transfer)
• Radiation ? Transfer of Electro- and Magnetic
• Conduction ? Transfer by molecular motions
  (Contact)
• Convection ? Transfer by turbulent mixing
  (parcels, bulk transport, advection)
• Latent (phase changes) ? "hidden heat"

124
Day 11

• Relationships between our state variables, r,
  a, T, and P.
• Boyles law
• Robert (Bob) Boyle (1600) said
• if T is constant, then

125
Day 11

• P1 x V1 P2 x V2 - or
• P x Volume Constant
• Recall, this is possible since Torricelli (1543)
  invented the barometer!
• So as a result of Boyles Law

126
Day 11

• As P increases, Volume decreases
• ? ?

127
Day 11

• Or as Volume increase, then P decreases
• ? ?

128
Day 11

• Charless Law
• Jaques Charles (1787) but stated formally J.
  Gay-Lussac (1802)
• (1787 Bonus question what else important
  happened in Sept. 1787?)
• Jack said if pressure is kept constant then
• Volume1/Volume2 Temperature1/Temperature2

129
Day 11

• Or
• Volume/Temperature Constant
• Thus, when gas is heated, volume goes up, when
  gas is cooled, volume decreases!

130
Day 11/12

• Combined or Ideal gas law
• Constant Pressure x Volume / Temperature
• Derivation of Ideal Gas law for Atmosphere
• ? Nee Avagadros hypothesis (derived
  experimentally, and derivable from kinetic theory
  of gasses)

131
Day 12

• His hypothesis Different gasses, each containing
  the same number of molecules, occupy the same
  volume at the same temperature and pressure.
• Kilogram molecular weight a kmol of material is
  its molecular weight expressed in kg. Thus, one
  kmol of water is 18.016 kg!
• Number of molecules is Avagadros number (Av)
• 6.022x 1026

132
Day 12

• Universal gas constant
• ? if we have a mixture of gases, each with its
  own ideal gas law (which is what Daltons Law
  implies!)
• Po Volume(gas) m(gas)R(gas)To where gas
  1,2,3, etc.
• where m is the number of moles of each gas!

133
Day 12

• Divide each equation by ng where ng is weight of
  a kmol of gas.
• Po Vg/ng (mg/ng) RgTo
• If each sample consists of 1 mol they have same
  number of molecules. Avagadros hypotheses all
  have same Volume. For each gas we can write
• Po (V1/ng) / To Po (Vo/ng) / To

134
Day 12

• or Po(Vg/ng) / To ng mgRg
• Since mgRg will be the same by Avagadros
  hypothesis
• Then mgRg R (Universal Gas constant)
• R 8314.3 J K-1 kmol-1

135
Day 12

• Thus, we must find the apparent weight of air, or
  take a weighted average of the gasses per mol.
• Molecular weight 28.97 kg / kmol n
• So then Pa R/n T
• Pa RT
• Or
• P rRT (R is a constant depending
  on individual Gas)

136
Day 12

• ? Or to include the effects of moisture
• P r Rd Tv (Rd 287.04 J K-1 kg-1)
• ? So lets go back to
• Po (Vg/ng) / To mg/ng R
• Can alternatively express RHS as R/ng R

137
Day 12

• Thus in ideal gas law we must express as
• PV R/n (air) T where air 28.97
• V is volume, where Volume could be anything. Well
  well specify some specific volume. (a
  vol/unit mass) which equals 1 kg. This is still
  volume, we are not changing variables here or
  playing fast and loose with the math.
• Thus
• Pa R/ (n air) T RdT
• (- or - P r R/n T rRdT)

138
Day 12

• Thats for dry air. In accounting for moisture
  ng 18.016. The amount of water is very
  variable, and we could be specifying a different
  R for air every time amount of moisture changes.
• ?Thats where concept of virtual temperature
  comes in. Thus,
• P a Rd Tv -or- P r Rd Tv

139
Day 12

• Q Which is more dense, dry air or moist air?
• Ever heard a baseball announcer talk about the
  ball carrying on a warm humid night?
• Dry air weights 28.97, throw in 18.016 for moist
  air.

140
Day 12

• In dry air, 1.00 28.97 for dry air 28.97
• Now, say the air was 4 vapor
• -age x mol. wt. Contribution
• 0.96 28.97 27.80
  
• 0.04 18.016 0.72
• ______________________________
• 28.53 (molecular weightof air vapor)

141
Day 12/13

• Application Lets derive expression for Virtual
  temperature (Wallace and Hobbs p51 52)
• Ideal Gas Laws, for water vapor and dry air

142
Day 13

• And use Daltons Law
• P Pd e
• r rd rv (1)
• so, substitute ideal gas laws into (1) to get
• Pd / (Rd T) (e / (Rv T))

143
Day 13

• And then
• (P e) / (Rd T) (e / (Rv T))
• use strategic multiplications of each term by
  1
• 1st term multiply by P / P
• 2nd term multiply by P Rd / P Rd

144
Day 13

• then well have a common factor to pull out (P
  / (Rd T))
• and we get
• Then rearrange

145
Day 13

• And finally

146
Day 13

• Celsius (1742) Temperature scale (Centigrade)
• Define at P Po 1000 mb at a state of thermal
  equilibruim
• Pure Ice and water mixture ? temp 0o C
• Pure water and steam mixture at equilibrium ? 100
  o C

147
Day 13

• If P Po alpha varies linearly with temp. Thus
• Y mx b
• T in oC is the slope
• T (Celsius) (at ao) / (a100 ao) 100
• This is the defining expression for Centigrade
  scale!

148
Day 13

• The Absolute or Kelvin Temp Scale
• T oC 100 at / (a100 ao) 100ao / (a100
  ao)
• VRBL CONST
• ? Or we can use this relationship to re-define
  the temperature scale by extrapolating to the
  point where all molecular motion stops and
  Specific Volume goes to 0!
• T (Absolute) VRBL T oC CONST

149
Day 13

• CONST 273.16
• So,
• K C 273.16
• Show Absolute zero -273.16

150
Day 13

• Solve for (a) at at
• at ao T oC / (a100 ao) 100
• at ao (1 1 / 273.16ToC)
• so if at 0 (or all molecular motion ceases),

151
Day 13

• then 11/273.16 ToC 0
• ? Solve T oC -273.16
• and then 273.16 oC 0 Absolute
• or 0 K (Volume of Ideal gas goes to 0)!!

152
Day 14

• The work done by an expanding gas
• Lets draw a piston

153
Day 13

• ? Consider a mass of gas at Pressure P in a
  cylinder of Cross section A
• Now, Recall from Calc III or Physics
• Work force x distance or Work Force dot
  distance
• So only forces parallel to the distance travelled
  do work!

154
Day 13

• Then,

155
Day 13

• But, we know that
• Pressure Force / Unit Area
• So then,
• Force P x Area

156
Day 13

• Total work increment now
• Well,
• Area x length Volume
• soo A ds dVol

157
Day 13

• Then we get the result
• ? Work
• Lets work with Work per unit mass
• ? Thus, we can start out with volume of only one
  1 kg of gas!!!

158
Day 13
159
Day 13
160
Day 13